Holographic characterization of irregular particles

ABSTRACT

Holographic Video Microscopy analysis of non-spherical particles is disclosed herein. Properties of the particles are determined by application of light scattering theory to holography data. Effective sphere theory is applied to provide information regarding the reflective index of a sphere that includes a target particle. Known particles may be co-dispersed with unknown particles in a medium and the holographic video microscopy is used to determine properties, such as porosity, of the unknown particles.

CROSS-REFERENCE TO RELATED APPLICATIONS

The present application is a continuation of U.S. Pat. App. No. 17/079,167, filed Oct. 23, 2020, now U.S. Pat. No. 11,543,338, which claims the benefit of and priority to U.S. Provisional Pat. App. No. 62/926,092 filed Oct. 25, 2019, both of which are incorporated by reference herein in their entireties.

TECHNICAL FIELD

The present disclosure relates generally to material analysis; more specifically to analysis of spherical particle properties and non-spherical particle properties by holographic video microscopy.

BACKGROUND

Holographic video microscopy (“HVM”) was originally developed for analyzing spherical particles composed of homogeneous media (see, e.g., Lee, et al., “Characterizing and tracking single colloidal particles wit video holographic microscopy,” Optics Express 15(26), pp. 18275-18282 (2007), incorporated herein by reference). HVM systems are described in U.S. Pat. Nos. 9,810,894 and 8,791,985, incorporated herein by reference. HVM is a holographic particle characterization that uses predictions by light scattering theory to analyze holographic snapshots of individual particles or objects. The hologram is generated by interaction of a light source, typically a collimated light source (e.g., a laser), and image acquisition by a microscope coupled with a videography device. The interaction of the light with the particle results in a scattering pattern. The scattering pattern is recorded as a hologram. The hologram can be analyzed to identify properties of the particle, such as the particle’s three-dimensional position, its radius, and its refractive index. A typical hologram subtends a 200×200 pixel array, with each pixel having a relative noise figure of 0.009, as determined by the median-absolute-deviation (“MAD”) metric. This technique has been shown to work reliably for colloidal spheres ranging in radius from 400 nm to 4 µm.

While HVM was originally focused on the detection and characterization of spherical particles, a wide range of non-spherical materials exist that would benefit from the ability to utilize HVM, such as non-spherical materials (e.g., 2-D flakes), organic molecules (e.g., proteins), agglomerated or flocculated materials, or porous particles. However, the mathematical requirements for applying a similar fitting of a light theory for non-spherical particles has proven overly complicated and computationally taxing such that it presents a barrier for use of HVM with such particles. Further, these more complex materials present additional properties that would benefit from an analytical technique for probing them (e.g., determining porosity properties).

Thus, there is a need for a process and systems to enable HVM analysis with non-spherical materials and to probe such materials unique properties.

SUMMARY

The in-line hologram of a micrometer-scale colloidal sphere can be analyzed with the Lorenz-Mie theory of light scattering to obtain precise measurements of the sphere’s diameter and refractive index. The same technique also can be used to characterize porous and irregularly shaped colloidal particles provided that the extracted parameters are interpreted with effective-medium theory to represent the properties of an equivalent effective sphere. In one embodiment described herein, the effective-sphere model consistently accounts for changes in the refractive index of the medium as it fills the pores of porous particles and therefore yields quantitative information about such particles’ structure and composition. One embodiment provides these capabilities through measurements on mesoporous spheres, fractal protein aggregates and irregular nanoparticle agglomerates, all of which are noteworthy for their industrial significance.

It should be appreciated that all combinations of the foregoing concepts and additional concepts discussed in greater detail below (provided such concepts are not mutually inconsistent) are contemplated as being part of the subject matter disclosed herein. In particular, all combinations of claimed subject matter appearing at the end of this disclosure are contemplated as being part of the subject matter disclosed herein.

BRIEF DESCRIPTION OF THE DRAWINGS

The foregoing and other features of the present disclosure will become more fully apparent from the following description and appended claims, taken in conjunction with the accompanying drawings. Understanding that these drawings depict only several implementations in accordance with the disclosure and are therefore not to be considered limiting of its scope, the disclosure will be described with additional specificity and detail through use of the accompanying drawings.

FIGS. 1A-1B show effective-sphere model for porous spheres (FIG. 1A) and irregular clusters (FIG. 1B). The medium of refractive index n_(m) fills the pores of a particle whose intrinsic refractive index is n₀. The effective sphere has refractive index n_(p) intermediate between n_(m) and n₀ and diameter d_(p).

FIG. 2 shows the distribution of particles’ diameters, d_(p) and refractive indexes, n_(p), with each point representing the holographically measured properties of a single particle. This sample is composed of monodisperse polystyrene spheres (n_(p) ≈ 1.6) and mesoporous silica spheres co-dispersed in a mixture of 90% 2,2′-thiodiethanol (“TDE”) in water. The two projections show the probability distribution of particle diameters and refractive indexes, the latter permitting clear differentiation between particle types. Horizontal dashed lines show the refractive indexes of fused silica and 90% TDE solution.

FIG. 3A shows the dependence of the population-averaged diameter, d_(p)(n_(m)), and FIG. 3B shows the refractive index, n_(p)(n_(m)), of the polystyrene and mesoporous silica spheres as a function of the refractive index of the medium, n_(m). Neither the particles’ diameters nor the measured refractive indexes of the polystyrene reference particles vary significantly with n_(m). The refractive index of the mesoporous silica spheres depends on n_(m) in agreement with Eq. 6. Error bars reflect population standard deviations for the two types of particles. Shaded boxes identify data from FIG. 2 .

FIG. 4A shows the joint distribution of particle diameter and refractive index for a mixture of IgG aggregates and silicone oil emulsion droplets in a sucrose solution at refractive index n_(m) = 1.429. Each analyzed particle is represented by a plot symbol, colored by the density of measurements, p(d_(p), n_(p)). The horizontal dashed line represents n_(m). The two types of particles cannot be distinguished in the projected distribution of particle diameters, p(d_(p)), but are clearly resolved in the distribution of refractive indexes, p(n_(p)). FIG. 4B shows the projected refractive index distributions as a function of the medium’s refractive index, n_(m). Curves are offset by 20 RIU for clarity. FIG. 4C shows the dependence of the mode refractive indexes for IgG and silicone particles as a function of n_(m). The horizontal dashed line represents the bulk refractive index of silicone oil, 1.404 ± 0.002. The diagonal curve is a fit to Eq. 6.

FIG. 5A shows holographic characterization data for protein aggregates dispersed in aqueous glycerol solutions at four different concentrations. Horizontal dashed lines indicate the refractive index of the medium, n_(m). FIG. 5B shows agglomerates of silica nanoparticles dispersed in a nanoparticle slurry whose refractive index and viscosity is adjusted with four different concentrations of glycerol.

FIG. 6A shows the width of the refractive index distribution, Δn_(p)(n_(m)), for mesoporous silica spheres dispersed in aqueous TDE solution spheres as a function of the medium’s refractive index, n_(m). FIG. 6B shows Δn_(p)(n_(m)) for protein aggregates in buffer with added sucrose, from FIGS. 4A-4C. Widths are unreliable in the shaded region where the distribution of protein aggregates overlaps with the distribution of silicone oil droplets. Dashed curves are fits to Eq. 7.

FIG. 7 illustrates a computer system for use with certain implementations.

Reference is made to the accompanying drawings throughout the following detailed description. In the drawings, similar symbols typically identify similar components, unless context dictates otherwise. The illustrative implementations described in the detailed description, drawings, and claims are not meant to be limiting. Other implementations may be utilized, and other changes may be made, without departing from the spirit or scope of the subject matter presented here. It will be readily understood that the aspects of the present disclosure, as generally described herein, and illustrated in the figures, can be arranged, substituted, combined, and designed in a wide variety of different configurations, all of which are explicitly contemplated and made part of this disclosure.

DETAILED DESCRIPTION

Holographic particle characterization is a high-throughput, low-cost technology that uses in-line holographic video microscopy to measure the diameters and refractive indexes of particles in their native media while simultaneously tracking their three-dimensional motions. It is compatible with any microfluidic system that provides access for optical microscopy. The measurement involves fitting a recorded hologram, pixel-by-pixel, to a model based on the Lorenz-Mie theory of light scattering by spheres. Colloidal particles with more complicated non-spherical materials can be analyzed by generalizing the light-scattering model at the cost of additional computational complexity.

Embodiments described herein relate generally to HVM, including HVM as applied to non-spherical particles (which may be solid or porous) and porous spherical particles (“irregular particles”). U.S. Pat. App. No. 16/076,265, incorporated herein by reference, described systems and methods that utilize a determination of the properties of an effective sphere, one that includes the non-spherical material and the surrounding and interstitial fluid medium. The Lorenz-Mie model for spheres also can be used to analyze these more general particles. The extracted particle characteristics can be interpreted with effective-medium theory to describe an effective sphere, shown schematically in FIGS. 1A-1B, whose boundary encloses the actual particle and whose properties represent contributions from both the particle itself and also the medium filling its pores. This approach has been demonstrated experimentally through measurements on porous colloidal spheres, dimpled spheres, fractal clusters of silica nanoparticles, protein aggregates and nanoparticle agglomerates. It has been validated by analyzing simulated holograms of dimpled spheres and fractal clusters. Thus, an “effective sphere” model can be used to determine properties of irregular particles. Dimpled spheres and fractal clusters are examples of the more general class of non-ideal particles whose physical properties can be assessed by applying the Lorenz-Mie model together with the effective-sphere interpretation. These non-ideal particles can have properties drawn from the set of aspherical shape, inhomogeneous chemical composition or both. The one requirement for successful implementation is that the effective sphere should be no larger than the range of sphere diameters that can be usefully analyzed with the Lorenz-Mie model.

These previous studies all focused on the relationship between the particle’s internal structure and the measured effective-sphere characteristics in a medium of fixed refractive index. Described herein are systems and methods, supported by experimental data, which provide a complementary analysis of the medium’s role in determining the effective-sphere characteristics as well as extended analytics for identifying properties specific to the irregular particles. Experimental data below focuses on porous particles, specifically mesoporous silica spheres, as a “irregular” particle. As used herein, non-spherical includes but is not limited to protein aggregates with branched fractal structure and nanoparticle agglomerates dispersed in chemical-mechanical planarization (“CMP”) slurries. Non-spherical particles, as a class of suitable subject particles, also includes coated spheres that are inhomogeneous because the coatings have optical properties distinct from those of the underlying sphere. Such coatings may be formed of atomic layers, molecular layers, layers of nanoparticles, layers of bound virus particles, or any other material that can be caused to bind to the surface of the sphere. The class of aspherical particles amenable to this technique include small aggregates of micrometer-scale spheres. These are physically distinct from fractal aggregates of nanoparticles in that the component spheres are large enough to be analyzed by Lorenz-Mie analysis by themselves, and that their clusters contain a small number of component spheres. These model systems were chosen for their relevance to drug delivery and catalysis, biopharmaceutical development and manufacturing, and semiconductor processing, respectively. The results of this study not only validate the effective-sphere model but also illustrate how to interpret holographic characterization data for the kinds of complex, multicomponent colloidal dispersions that are encountered in real-world applications.

In one embodiment, HVM is carried out using a basic setup where the microscope illuminates the sample with a collimated laser beam at vacuum wavelength λ. This incident wave can be modeled as a monochromatic plane wave propagating along ẑ with linear polarization along x̂:

E₀(r) = u₀e^(ikz)x̂.

A small particle at r_(p) scatters a portion of this field,

E_(s)(r) = u₀e^(ikz_(p))f_(s)(k(r − r_(p))),

to position r in the focal plane of the microscope, where k = 2πn_(m)/λ is the wavenumber of the light in a medium of refractive index n_(m), and f_(s)(kr) is the Lorenz-Mie scattering function. The microscope magnifies the interference pattern formed by the incident and scattered fields, and relays it to a video camera. The recorded intensity is then divided by an image of the background illumination to obtain a normalized hologram that can be modeled as

b(r) = |x̂ + e^(ikz_(p))f_(s)(k(r − r_(p)))|².

Through the Lorenz-Mie function, this expression for b(r) is parameterized by the sphere’s diameter, d_(p), and its refractive index, n_(p), at the imaging wavelength. Fitting for these characterization data and the particle’s position requires three calibration parameters: the wavelength of the illumination, the magnification of the optical train, and the refractive index of the medium.

Single-particle characterization measurements are combined into population distributions. Each point in this scatter plot reflects the measured diameter and refractive index of one particle. In one embodiment, the points are colored by the probability density of measurements, ρ(d_(p), n_(p)), such as computed using a kernel density estimator. Clusters of points reflect distinct populations of particles in the sample.

Effective Sphere Model

The Lorenz-Mie function, f_(s)(kr), describes light scattering by an isotropic homogeneous sphere and is not inherently suitable for describing light scattering by porous, irregularly shaped or otherwise inhomogeneous particles. Generalizing f_(s)(kr) to accommodate more general particle shapes is feasible but is computationally costly. The efficiency of the standard Lorenz-Mie implementation is retained by treating irregular and inhomogeneous particles as if they were homogeneous spheres whose measured properties then can be interpreted as averages over the media contained within their least bounding spheres, as suggested schematically in FIGS. 1A-1B.

The basis for this effective-sphere model is provided by Maxwell Garnett effective medium theory, according to which a particle composed of N different phases dispersed in a medium of refractive index n_(m), has an effective refractive index, n_(p), that satisfies the condition

$L_{m}\left( n_{p} \right) = {\sum_{j = 1}^{N}{\phi_{j}L_{m}\left( n_{j} \right),}}$

where the Lorentz-Lorenz function is

$L_{m}(n) = \frac{n^{2} - n_{m}^{2}}{n^{2} + 2n_{m}^{2}},$

and where ϕ_(j) is the volume fraction of the j^(th) phase within the effective sphere. Eqs. 4a-4b reasonably describe the light-scattering properties of particles whose inhomogeneities are uniformly distributed when viewed on scales comparable to the wavelength of light.

Porous spheres and colloidal aggregates may be modeled as two-phase systems composed of a host material of refractive index n₀ at volume fraction ϕ whose pores are filled with the surrounding fluid medium, as shown schematically in FIGS. 1A-1B. Such a particle’s porosity is related to its volume fraction by p = 1 - ϕ. Noting that L_(m)(n_(m)) = 0, the effective-sphere model then predicts

L_(m)(n_(p)) = ϕL_(m)(n₀),

so that the effective sphere’s refractive index depends on that of the surrounding medium as

$n_{p}\left( n_{m} \right) = n_{m}\sqrt{\frac{1 - 2\phi L_{m}\left( n_{0} \right)}{1 + \phi L_{m}\left( n_{0} \right)}}.$

The effective sphere is index matched (n_(p) = n_(m)) in a medium that matches its host material, n_(m) = n₀. A non-porous sphere with ϕ = 1 has the refractive index of its material, n_(p) = n₀, as expected, and this value does not vary with the refractive index of the medium.

Polydispersity

The above discussion focused on how a porous particle’s effective refractive index depends on its porosity and the refractive index of the medium. As noted, irregular particles present a range of properties that can be probed and analyzed but for which traditional HVM has not provided a method. For example, porous particles have several different parameters associated with their pores, including pore diameter, pore volume, and pore interconnection. Thus, one can measure the polydispersity of porosity for the particles in a sample. The same approach can be used to assess the properties of porous particles created by aggregation of colloidal spheres. In addition to being porous, such consolidated particles also are irregularly shaped.

The range, Δn_(p), of apparent refractive indexes for clusters of a given size presumably reflects variations in the clusters’ structures and therefore the spread, Δϕ, in values of ϕ. Eq. 6 accounts for the dependence of Δn_(p) on Δϕ through:

$\Delta n_{p} = \left| \frac{\partial n_{p}}{\partial\varnothing} \right|\Delta\varnothing\text{=}\frac{3}{2}\frac{n_{m}^{2}}{n_{p}}\frac{\left| {L_{m}\left( n_{0} \right)} \right|}{\left\lbrack {1 + \varnothing L_{m}\left( n_{0} \right)} \right\rbrack^{2}}\Delta\varnothing.$

Most notably, this result shows that p(n_(p)) narrows as the refractive index of the medium approaches that of the monomers because L_(m)(n_(p)) = 0.

The ability, or inability, of probe particles, such as dissolved molecules or co-dispersed nanoparticles to permeate the particles’ pores provides information about the size and connectivity of the particles’ pores. One embodiment of an HVM system and process provides information regarding such. An unknown target particles may be analyzed in a medium co-dispersed with probe particles. The probe particle have known properties such as refractive index and diameter in some embodiments. Likewise, the medium may have known properties such as refractive index. The refractive index of the dispersed probe particles in the medium may also be known. One or more of the target particle are dispersed in the medium with the probe particles. Holographic characterization is done as discussed above, including the use of an effective sphere model. The refractive index of the effective sphere will comprise the refractive index of the target particle and the medium and, if the probe particles are able to enter the pores of the target particle, the effective sphere’s refractive index will also reflect the presence of such probe particles. This process may be performed with various probe particle, such as varying their known diameter, to determine properties such as diameter of the pores of the target particles by identifying when the probe particles are within the pores (as reflected in the effective sphere refractive index) or not. In one embodiment, the target particles are spherical and the holographic characterization applies Lorenz-Mie theory but does not need to utilize the effective sphere analysis since the particles are spherical.

For example, if the particles’ effective refractive index tracks the concentration of molecules, then one can conclude that the molecules can permeate the pores at their bulk concentration. In that case, it can be concluded that that the pores are large enough to accommodate the molecules and are connected in such a way that the molecules can find their way in. If, conversely, the particles’ effective refractive index does not track the concentration of molecules, it can be concluded that the pores cannot accommodate the molecules, either because the pores are too small, because they are inaccessible or because they are chemically inhospitable. Further information about the pore structure, functionality and accessibility can be obtained by repeating these measurements with different molecules of different sizes and different chemical properties. Thus, one embodiment relates to a screening method for providing HVM tests on samples with different known properties to determine the impact on the results and identify populations of particles that are change and those that are not changing with the changes in the tested sample, such as by adjusting the media/solvent’s refractive index.

Further, just as dissolved molecules of known characteristics can be used as probe particles to probe the unknown pore structure of a target particle, well-characterized porous spheres can be used to assess the properties of unknown species in the fluid medium. The refractive index of the medium can be changed by adding a miscible fluid of known properties. If the effective refractive index of the sampled known porous spheres tracks (i.e., both increase, decrease, or stay the same, including, in some embodiments, the extent of change) the concentration of the unknown fluid with particles to be identified, then any molecules (nanoparticles) in the unknown fluid must be able to permeate the pores. If, conversely, the effective refractive index does not track the concentration of the unknown fluid, then that fluid contains molecules (or nanoparticles) that cannot permeate the pores. Repeating this measurement with different types of porous particles can help to identify the nature of the molecules (nanoparticles) in the unknown fluid. Multiple classes of porous particles can be combined to perform multiple variants of this test in parallel, the different classes of particles being distinguished by size, or by other characteristics such as the refractive index of the underlying material. Such aspect shall be referred to as holographic perfusion chromatography.

Experimental Results Holographic Particle Characterization

The data for holographic particle characterization are acquired with in-line HVM. For the experimental setup, data acquisition and analysis were performed with a Spheryx xSight® device. Samples are loaded into xSight® in disposable xCell microfluidic sample chips, each of whose reservoirs holds 30 µL. xSight® mixes the sample and then transfers 3 µL through its holographic microscope using pressure gradients. The chip’s observation volume has an optical path length of 50 µm and provides the microscope with a clear 150 µm ×120 µm field of view, given the microscope’s magnification of 120 nm/pixel. The instrument records holograms at λ = 447 nm and can analyze particles ranging in size from d_(p) = 400 nm to d_(p) = 10 µm. A typical 15 min measurement can analyze particle concentrations as low as 10³ particles/mL and as high as 10⁷ particles/mL. The lower limit is set by counting statistics. The upper limit is set by the need to minimize overlap between holograms of multiple particles in the camera’s field of view.

Single-particle characterization measurements are combined into population distributions such as the example in FIG. 2 . Clusters of points reflect distinct populations of particles in the colloidal sample. In the case of FIG. 2 , two populations are clearly distinguishable by their differing refractive indexes even though their size distributions overlap. The points are colored by the probability density of measurements, p(d_(p), n_(p)), computed using a kernel density estimator.

Results & Discussion Effective-Sphere Characterization of Mesoporous Silica Spheres

The effective-sphere model’s predictions are tested by measuring the properties of well-characterized porous particles dispersed in media with a range of refractive indexes. The particles used for this study are nominally 2.5 µm-diameter mesoporous silica spheres with 4 nm-diameter pores (Sigma-Aldrich, catalog number 806951). These test particles are co-dispersed with 2.5 µm-diameter cross-linked polystyrene spheres (Spherotech, catalog number PP10-20-10), which serve as a control because they are not porous and should not respond in any way to changes in the properties of the medium.

Mesoporous silica spheres and polystyrene controls are dispersed at a total concentration of 10⁶ particles/mL in mixtures of deionized water and TDE (Sigma-Aldrich catalog number 166782, CAS No. 111-48-8). TDE is miscible with water and has a refractive index of 1.520 at the imaging wavelength, which substantially exceeds the value for fully dense fused silica, n₀ = 1.466.

The data in FIG. 2 were obtained with this system at 90% TDE by volume, which has a holographically-measured refractive index of n_(m) = 1.510 ± 0.007. The scatter plot shows results for 894 polystyrene spheres and 352 silica spheres. FIGS. 3A-3B summarize results from ten such data sets over the range from pure water (n_(m) = 1.339 ± 0.001) to 90% TDE. As anticipated, the measured properties of the polystyrene control particles (yellow squares) do not depend on the refractive index of the medium. Both the mean diameter of these spheres, d_(p) = (2.55 ± 0.04) µm, and the refractive index, n_(p) = 1.603 ± 0.005, are consistent with the manufacturer’s specification over the entire range of n_(m).

The measured diameter of the mesoporous silica spheres also is insensitive to changes in n_(m). The mean refractive index, by contrast, increases from n_(p) = 1.339 ± 0.001 in deionized water to n_(p) = 1.482 ± 0.001 in 90% TDE. The dashed curve through the refractive index data in FIG. 3B is a fit to Eq. 6 that tracks this trend and yields n₀ = 1.455 ± 0.001 and ϕ = 0.46 ± 0.01.

The 0.7% discrepancy between n₀ and the refractive index of fused silica may be ascribed in part to the well-documented difference in density between emulsion-polymerized silica and fused silica. The discrepancy also is likely to depend on the molecules’ sizes and their affinity for silica, both of which affect their ability to access the particles’ pores. Pores that are inaccessible to the high-index species in solution will tend to reduce a sphere’s apparent porosity. The inaccessible volume being filled with low-index solvent, this effect also tends to reduce the apparent refractive index of the silica matrix.

Differences in accessibility may explain the subtle species-dependent variations in n₀ and porosity reported in Table 1. In addition to the results obtained with TDE, this table summarizes two additional series of measurements using glycerol (Sigma-Aldrich, catalog number G9012, CAS 56-81-5, refractive index 1.526 ± 0.002) and saturated sucrose solution (Sigma-Aldrich, catalog number S8501, CAS 57-50-1, refractive index 1.501 ± 0.002) to tune the refractive index of the aqueous medium. Both yield slightly smaller values for n₀ and p than TDE, and in both cases the differences are statistically significant. Sucrose is substantially bulkier than TDE, which suggests that the difference might be attributed to steric exclusion. Glycerol is comparable in size to TDE but nevertheless yields smaller values for n₀ and p. The difference in this case might reflect differences in the solute molecules’ interactions with silica.

TABLE 1 Effective-sphere parameters for mesoporous silica spheres in media with varying refractive indexes. Specified high-index species are added to the aqueous medium to adjust the refractive index. Fitting the dependence of n_(p)(nm) to Eq. 6 yields the refractive index of the sphere’s matrix, n₀, and the spheres’ mean porosity, p High-Index Species n₀ p TDE 1.455 ± 0.001 0.54 ± 0.01 glycerol 1.448 ± 0.001 0.51 ± 0.01 sucrose 1.444 ± 0.001 0.52 ± 0.01

Broadly speaking, all three results suggest that the mesoporous silica spheres have a mean porosity exceeding 50% in a matrix whose optical properties are consistent with low-density silica. Consistency among these results serves to validate the effective-sphere model’s predictions for porous spheres. The subtle but significant differences in results obtained with different high-index species suggest that holographic porosimetry based on solvent perfusion may provide useful insights into pore structure and functionality.

Effective-Sphere Analysis of Protein Aggregates and Nanoparticle Agglomerates

Having successfully applied the effective-sphere model to mesoporous spheres, this is used to interpret holographic characterization data for irregularly shaped particles. FIG. 4A presents holographic characterization data for a mixture of protein aggregates and silicone oil emulsion droplets. The two populations cannot be distinguished by size, but are clearly differentiated by refractive index.

The aggregates in this sample are composed of human immunoglobulin G (“IgG”) (Sigma-Aldrich catalog number 12511, MDL number MFCD00163923), dissolved in Tris buffer (Sigma-Aldrich, catalog number 648314, CAS number 77-86-1) at a concentration of 5 mg mL⁻¹. IgG is a protein that naturally tends to aggregate into branched fractal clusters. Holograms created by such clusters can be analyzed with the effective sphere model, as indicated schematically in FIG. 1B, to obtain estimates for each cluster’s effective diameter and refractive index.

Emulsion droplets are created by manually agitating silicone oil (Sigma-Aldrich, product number 378399, CAS number 63148-62-9) in water. The emulsion then is blended into the protein solution at a concentration of 10⁵ droplets/mL. Silicone oil droplets are common contaminants in biopharmaceutical products. In the present application, they serve as non-porous reference spheres.

Adding saturated sucrose solution to the buffer increases its refractive index. FIG. 4B shows how the measured distribution of refractive indexes, p(n_(p)), depends on sucrose concentration through its influence on n_(m). The data in FIG. 4A were obtained for a sample at n_(m) = 1.429 ± 0.002 and are reproduced in FIG. 4B. One peak in the bimodal distribution remains centered at refractive index of 1.404 ± 0.002, which is consistent with the refractive index of bulk silicone oil. The other peak tracks changes in n_(m) as anticipated by the effective sphere model. The former has been identified as the contribution of silicone oil droplets and the latter as reflecting the properties of protein aggregates.

The data in FIG. 4C show how the aggregates’ mode refractive index, n_(p) (n_(m)), responds to changes in n_(m). The diagonal dashed curve is a fit to Eq. 6 that yields an effective volume fraction of ϕ = 0.03 ± 0.03, which corresponds to a porosity of ρ = 0.97 ± 0.03. Such a high porosity is expected for fractal aggregates that have grown to many times the size of their monomers.

The effective-sphere model implicitly treats the aggregates as homogeneously porous particles whose internal structure is independent of size and buffer composition. Accounting for the fractal aggregates’ size-dependent porosity does not change the magnitude of the observed porosity in the experimentally accessible size range. More importantly, the results presented in FIGS. 4A-4C offer insights into the particles’ composition without requiring assumptions or a priori knowledge of their detailed structure.

FIG. 5A shows analogous characterization data for IgG aggregates when glycerol is used to adjust n_(m) instead of sucrose. Results are presented for glycerol at 0%, 30%, 50%, and 60% by volume. The buffer’s refractive index at each concentration is indicated by a horizontal dashed line. As with the sucrose data, the aggregates’ measured refractive indexes track n_(m). The mean porosity inferred from the mode values of n_(p)(nm) is ρ = 0.97 ± 0.03, which also is consistent with results obtained with sucrose. This is not to say that the aggregates have the same structure in the different media but that any structural changes are not apparent in the aggregates’ overall porosity.

Analogous results are plotted in FIG. 5B for nanoparticle agglomerates in a slurry of silica nanoparticles (General Engineering & Research 80 nm, SIO2-743). Unless stabilized by added surfactants, nanoparticles in this slurry also tend to agglomerate into fractal clusters. This system also is noteworthy because the population of dispersed nanoparticles render the slurry turbid. Holographic particle characterization previously has been shown to be effective for characterizing micrometer-scale particles in such media.

The effective refractive index of nanoparticle agglomerates tracks the refractive index of the medium, as anticipated by the effective sphere model. Also as expected, the size distribution of nanoparticle agglomerates appears not to vary appreciably with the addition of glycerol. It is believed that these agglomerates also are highly porous and that their pores are perfused by the fluid medium. Reproducing these trends in such physically distinct systems as protein aggregates and nanoparticle agglomerates lends further credibility to the effective-sphere interpretation of irregular clusters’ light-scattering properties.

Polydispersity of Porosity

The projected refractive index distributions in FIGS. 5A-5B not only shift upward as nm increases, but also become more narrow. The range, Δn_(p), of apparent refractive indexes for clusters of a given size presumably reflects variations in the clusters’ structures and therefore the spread, Δϕ, in values of ϕ.

FIG. 6A shows the result of applying this analysis, applying Eq. 6, to the data for mesoporous silica spheres dispersed in TDE. The width of the refractive index distribution at each value of nm is estimated with robust principal component analysis, as is the uncertainty of the width. The dashed curve in a one-parameter fit to Eq. 7 for Δϕ using the value of n₀= 1.455 obtained from n_(p)(n_(m)). The result, Δϕ = 0.023 ± 0.001, is consistent with a 4% polydispersity in these particles’ porosity.

Applying the same analysis to the data for protein aggregates in sucrose solution yields the results in FIG. 6B. The widths of the distributions cannot be assessed reliably in the range of n_(m) for which the silicone oil distribution overlaps with the aggregate distribution, which is indicated by the shaded region. Fitting the remainder of the data to Eq. 7 yields Δϕ = 0.05 ± 0.02 and n₀ = 1.575 ± 0.008. This value for the monomer refractive index is consistent with expectations for proteins such as IgG.

Conclusions

The experimental studies presented here demonstrate that the effective-sphere model usefully accounts for the properties of porous spheres and irregularly shaped colloidal particles as reported by Lorenz-Mie analysis of holographic microscopy data. Specifically, these studies validate the predicted role of the medium in establishing a porous particle’s effective refractive index. This dependence is characteristic of porous particles and can be used to differentiate them from non-porous particles, such as the polystyrene spheres and silicone oil droplets used as references in this study.

Fitting measurements of n_(p)(n_(m)) to Eq. 6 yields estimates for the particles’ porosity and the refractive index of their matrix. These characterization results are found to depend on the choice of compounds used to adjust the medium’s refractive index. Tracking this dependence may be useful for probing the size distribution, connectivity and surface functionality of the pores within porous particles.

Changes in the medium that affect the refractive index also influence other physical properties. The viscosity of the samples in FIGS. 3A-3B, for example, increases from 1 × 10⁻³ Pas in pure water to 6 × 10⁻³ Pas in 90% TDE. Consistent characterization results for polystyrene standards demonstrate that the approach to holographic particle characterization implemented in xSight® is insensitive to such ancillary effects.

The ability of holographic particle characterization to differentiate porous colloidal particles from non-porous particles has immediate applications for assessing the quality of protein-based biopharmaceutical products and nanoparticle-based CMP slurries used for semiconductor manufacturing. Monitoring solute perfusion in mesoporous particles may provide an approach to porosimetry that complements mercury intrusion, helium isotherms, and electron microscopy, with particular benefits for analyzing the pore structure of colloidal materials.

Definitions

As used herein, the singular forms “a,” “an,” and “the” include plural referents unless the context clearly dictates otherwise. Thus, for example, the term “a member” is intended to mean a single member or a combination of members, “a material” is intended to mean one or more materials, or a combination thereof.

As used herein, the terms “about” and “approximately” generally mean plus or minus 10% of the stated value. For example, about 0.5 would include 0.45 and 0.55, about 10 would include 9 to 11, about 1000 would include 900 to 1100.

It should be noted that the term “exemplary” as used herein to describe various embodiments is intended to indicate that such embodiments are possible examples, representations, and/or illustrations of possible embodiments (and such term is not intended to connote that such embodiments are necessarily extraordinary or superlative examples).

As used herein, the terms “coupled,” “connected,” and the like mean the joining of two additional intermediate members being integrally formed as a single unitary body with one another or with the two members or the two members and any additional intermediate members being attached to one another.

As shown in FIG. 7 , e.g., a computer-accessible medium 120 (e.g., as described herein, a storage members directly or indirectly to one another. Such joining may be stationary (e.g., permanent) or moveable (e.g., removable or releasable). Such joining may be achieved with the two members or the two members and any device such as a hard disk, floppy disk, memory stick, CD-ROM, RAM, ROM, etc., or a collection thereof) can be provided (e.g., in communication with the processing arrangement 110). The computer-accessible medium 120 may be a non-transitory computer-accessible medium. The computer-accessible medium 120 can contain executable instructions 130 thereon. In addition or alternatively, a storage arrangement 140 can be provided separately from the computer-accessible medium 120, which can provide the instructions to the processing arrangement 110 so as to configure the processing arrangement to execute certain exemplary procedures, processes and methods, as described herein, for example. The instructions may include a plurality of sets of instructions.

System 100 may also include a display or output device, an input device such as a keyboard, mouse, touch screen or other input device, and may be connected to additional systems via a logical network. Many of the embodiments described herein may be practiced in a networked environment using logical connections to one or more remote computers having processors. Logical connections may include a local area network (LAN) and a wide area network (WAN) that are presented here by way of example and not limitation. Such networking environments are commonplace in office-wide or enterprise-wide computer networks, intranets and the Internet and may use a wide variety of different communication protocols. Those skilled in the art can appreciate that such network computing environments can typically encompass many types of computer system configurations, including personal computers, hand-held devices, multiprocessor systems, microprocessor-based or programmable consumer electronics, network PCs, minicomputers, mainframe computers, and the like. Embodiments of the invention may also be practiced in distributed computing environments where tasks are performed by local and remote processing devices that are linked (either by hardwired links, wireless links, or by a combination of hardwired or wireless links) through a communications network. In a distributed computing environment, program modules may be located in both local and remote memory storage devices.

Various embodiments are described in the general context of method steps, which may be implemented in one embodiment by a program product including computer-executable instructions, such as program code, executed by computers in networked environments. Generally, program modules include routines, programs, objects, components, data structures, etc. that perform particular tasks or implement particular abstract data types. Computer-executable instructions, associated data structures, and program modules represent examples of program code for executing steps of the methods disclosed herein. The particular sequence of such executable instructions or associated data structures represents examples of corresponding acts for implementing the functions described in such steps.

Software and web implementations of the present invention could be accomplished with standard programming techniques with rule based logic and other logic to accomplish the various database searching steps, correlation steps, comparison steps and decision steps. It should also be noted that the words “component” and “module,” as used herein and in the claims, are intended to encompass implementations using one or more lines of software code, and/or hardware implementations, and/or equipment for receiving manual inputs.

It is important to note that the construction and arrangement of the various exemplary embodiments are illustrative only. Although only a few embodiments have been described in detail in this disclosure, those skilled in the art who review this disclosure will readily appreciate that many modifications are possible (e.g., variations in sizes, dimensions, structures, shapes and proportions of the various elements, values of parameters, mounting arrangements, use of materials, colors, orientations, etc.) without materially departing from the novel teachings and advantages of the subject matter described herein. Other substitutions, modifications, changes and omissions may also be made in the design, operating conditions and arrangement of the various exemplary embodiments without departing from the scope of the present invention.

While this specification contains many specific implementation details, these should not be construed as limitations on the scope of any inventions or of what may be claimed, but rather as descriptions of features specific to particular implementations of particular inventions. Certain features described in this specification in the context of separate implementations can also be implemented in combination in a single implementation. Conversely, various features described in the context of a single implementation can also be implemented in multiple implementations separately or in any suitable subcombination. Moreover, although features may be described above as acting in certain combinations and even initially claimed as such, one or more features from a claimed combination can in some cases be excised from the combination, and the claimed combination may be directed to a subcombination or variation of a subcombination. 

What is claimed is:
 1. A method of characterizing the porosity of colloidal particles dispersed in a fluid medium in a sample, comprising: flowing the sample through an observation volume of a holographic microscope; generating a first holographic image based upon holographic video microscopy of the sample within the observation volume at a first time; analyzing the first holographic image for one or more regions of interest corresponding to a porous particle of interest; normalizing the region of interest for a contribution of a wave created by interaction of light with the sample; fitting the normalized region of interest to a light scattering theory; and characterizing one or more properties of pores of the particle of interest.
 2. The method of claim 1, wherein the one or more properties of the pores include at least one of pore diameter, pore volume, and more interconnection.
 3. The method of claim 1, wherein the porous particle of interest comprises a plurality of aggregated colloidal spheres.
 4. The method of claim 1, wherein the porous particle of interest comprises a particle with a coating.
 5. The method of claim 1, wherein fitting the normalized region of interest to a light scattering theory includes applying an effective sphere model through a Lorenz-Mie function to determine the reflective index of an effective sphere surrounding the porous particle of interest.
 6. A method of characterizing the porosity of colloidal particles dispersed in a fluid medium in a sample, comprising: obtaining first sample data by: flowing a first sample with a medium having a first refractive index and having a plurality of particles including a first particle type and a second particle type, through an observation volume of a holographic microscope, generating a first holographic image based upon holographic video microscopy of the first sample within the observation volume at a first time, analyzing the first holographic image for first sample regions of interest corresponding to at least one of the first set of particles and at least one of the second set of particles, and fitting the first sample regions of interest to a light scattering theory; obtaining second sample data by: flowing a second sample with a second medium with a different refractive index than the first medium, having a plurality of particles including the first particle type and the second particle type, through the observation volume of the holographic microscope, generating a second holographic image based upon holographic video microscopy of the second sample within the observation volume at a second time, analyzing the second holographic image for second sample regions of interest corresponding to at least one of the first set of particles and at least one of the second set of particles, and fitting the second sample regions of interest to the light scattering theory; and determining one or more properties of pores of at least one of the first particle type and the second particle type.
 7. The method of claim 6, wherein the one or more properties of the pores include at least one of pore diameter, pore volume, and more interconnection.
 8. The method of claim 6, wherein the first particle type has a known radius and refractive index.
 9. The method of claim 8, wherein the first particle type has a known pore size.
 10. The method of claim 9, wherein determining the one or more properties of pores comprises determining one or more properties of pores of the second particle type.
 11. The method of claim 10, wherein fitting the first normalized region of interest to a light scattering theory includes applying an effective sphere model through a Lorenz-Mie function to determine a reflective index of an first effective sphere and wherein fitting the second normalized region of interest to the light scattering theory includes applying the effective sphere model through the Lorenz-Mie function to determine a reflective index of a second effective sphere.
 12. A method of analyzing a sample comprising: performing holographic characterization of a target particle of a group of target particles, the target particle co-dispersed with first probe particles in a first medium; determining a refractive index for an effective sphere encompassing the target particle; performing a second holographic characterization of a second target particle of the group of target particles, the second target particle co-dispersed with second probe particles in a second medium; determining a second refractive index for a second effective sphere encompassing the second target particle; characterizing one or more properties of the group of target particles based on a comparison of the first refractive index and the second refractive index.
 13. The method of claim 12, wherein the first medium and the second medium are the same.
 14. The method of claim 12, wherein the first probe particle, the first medium, the second probe particle, and the second medium each have a known refractive index.
 15. The method of claim 12, wherein the first probe particle and the second probe particle have different diameters. 